In the local-density approximation (LDA), the many-electron problem is approximated by a set of single-particle equations which are solved with the self-consistent field method. The total energy is minimized. The total energy is taken to be the sum of a kinetic energy, T, the classical Hartree term for the electron density, , the electron-nucleus energy, , and the exchange-correlation energy, , which takes into account approximately the fact that an electron does not interact with itself, and that electron correlation effects occur.
One solves the Kohn-Sham orbital equations
The charge density is given by
where the 2 accounts for double occupancy of each spatial orbital because of spin degeneracy. The potential, , is the external potential; in the atomic case, this is where Z is the atomic number. The exchange correlation potential, , is a function only of the charge density, i.e., . We use the functional of Vosko, Wilk, and Nusair (1980), as described above.
The various parts of the total energy are given by:
where is the exchange-correlation energy per particle for the uniform electron gas of density . This approximation for is the principal approximation of the LDA.